Advances in Applied Clifford Algebras

In this paper, the study of linear differential equations involving one conventional and two nilpotent variables is started. This is a natural extension of the case of one involved nilpotent (para-Grassmann) variable studied earlier. In the case considered here, the two nilpotent variables are assumed to commute, hence they are generators of a (generalized) zeon algebra. Using the natural para-supercovariant derivatives $$D_i$$ transferred from the study of a para-Grassmann variable, we consider linear differential equations of order at most two in $$D_i$$ and discuss the structure of their solutions. For this, convenient graphical representations in terms of simple graphs are introduced.

In this paper the mean ergodic theorem in bicomplex Banach modules is studied. Under appropriate conditions of boundness for the iterates compositions of a bicomplex linear and bounded operator on a bicomplex Banach module, and of weak compactness of the average sequence on the idempotent components, analogous to that of the classical case on Banach spaces, strong convergence of the mean sequence is achieved. Also, a result on ergodicity is given for bounded bicomplex strongly continuous semigroups in bicomplex Banach modules.

This work explains how three dimensional quadrics can be defined by the outer products of conformal geometric algebra points in higher dimensions. These multivector expressions code all types of quadrics in arbitrary scale, location and orientation. Furthermore, a newly modified (compared to Breuils et al. in Adv Appl Clifford Algebras 28(35):1–16, 2018. https://doi.org/10.1007/s00006-018-0851-1 ) approach now allows not only the use of the standard intersection operations, but also of versor operators (scaling, rotation, translation). The new algebraic form of the theory will be explained in detail.

In this paper we study pseudo $${\varphi}$$ -structure on a differential manifold in which is a tensor field of type (1, 1) satisfying $${\varphi^{3}\,-\,\varphi=0}$$ . We obtain basic curvature tensor fields for pseudo f-manifolds with complemented frame.

Trong bài báo này, chúng tôi mở rộng khái niệm nhóm Brauer–Clifford sang trường hợp đại số $$(S,{{\mathcal {G}}},H)$$, khi H là một đại số Hopf đồng giao hoán, $${{\mathcal {G}}}$$ là một đại số Lie trong thể loại monoidal đối xứng của các mô-đun bên trái H, và S là một đại số giao hoán, nó là một đại số mô-đun H, một đại số mô-đun $${\mathcal G}$$ và hành động H là tương thích với hành động $${\mathcal {G}}$$. Nhóm Brauer–Clifford này hóa ra là một ví dụ của nhóm Brauer trong một thể loại monoidal đối xứng.

We show that the binary representation of the integers plays a role in several parts of Clifford algebras.

A modification of the classical theory of spherical harmonics is presented. The space $${\mathbb {R}}^d = \{(x_1,\ldots ,x_d)\}$$ is replaced by the upper half space $${{\mathbb {R}}}_{+}^{d}=\left\{ (x_1,\ldots ,x_d), x_d > 0 \right\} $$, and the unit sphere $$S^{d-1}$$ in $${\mathbb {R}}^d$$ by the unit half sphere $$S_{+}^{d-1}=\left\{ (x_1,\ldots ,x_d): x_1^2 + \cdots + x_d^2 =1, x_d > 0 \right\} $$. Instead of the Laplace equation $$\Delta h = 0$$ we shall consider the Weinstein equation $$x_d\Delta u + (d-2)\frac{\partial u }{\partial x_d}= 0$$. The Euclidean scalar product for functions on $$S^{d-1}$$ will be replaced by a non-Euclidean one for functions on $$S_{+}^{d-1}$$. It will be shown that in this modified setting all major results from the theory of spherical harmonics stay valid. In case $$d=3$$ and $$d=4$$ the modified theory has already been treated.

Direct product rings have received relatively little attention, perhaps because they are sometimes labeled “trivial” [8, p.6]. Nevertheless, the 2-dimensional direct product ring of the reals, when expressed in the “hyperbolic basis”, is analogous in many ways to the system of complex numbers and also has a physical interpretation. This prompted an exploratory foray into the world ofn-dimensional direct product rings of the reals to see how much can be extended from the 2-dimensional case (see, e.g. [3,4,5]). Section 1 provides algebraic notation, up to the point of defining polar coordinates. Section 2 uses analysis to explore differentiability and conformality.